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1D Scattering through time dependent media with memory

Jeffrey Galkowski, Zhen Huang, Maciej Zworski

Abstract

We construct a scattering matrix with operator valued entries describing solutions to the 1+1 wave equation where permittivities has memory and depends on time and space. It is the analogue of the scattering matrix for spatially localised perturbations where the entries are functions of frequency and appear as Fourier multipliers in solutions of the wave equation. This provides a mathematical explanation of the numerical construction in the recent paper by Horsley et al. The appendix by Zhen Huang and Maciej Zworski presents a numerical scheme for solving the wave equation considered in this article.

1D Scattering through time dependent media with memory

Abstract

We construct a scattering matrix with operator valued entries describing solutions to the 1+1 wave equation where permittivities has memory and depends on time and space. It is the analogue of the scattering matrix for spatially localised perturbations where the entries are functions of frequency and appear as Fourier multipliers in solutions of the wave equation. This provides a mathematical explanation of the numerical construction in the recent paper by Horsley et al. The appendix by Zhen Huang and Maciej Zworski presents a numerical scheme for solving the wave equation considered in this article.
Paper Structure (15 sections, 10 theorems, 158 equations, 2 figures)

This paper contains 15 sections, 10 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scattering for permittivities with memory
  3. The wave equation
  4. General theory
  5. An abstract scattering operator
  6. A class of permittivities
  7. Existence and uniqueness
  8. An operator valued scattering matrix
  9. Review of standard scattering matrix
  10. Construction of $u( x, \omega )$ in Theorem \ref{['t:1']}
  11. Non-existence of purely outgoing solutions
  12. Proof of Theorem 2
  13. Mathematical setup
  14. Numerical schemes
  15. Matlab codes

Key Result

Theorem 1

Suppose that assumptions in eq:newave hold and $f \in ( \omega + i )^{-\textcolor{blue}{1}} H_{-R }$. Then there exist bounded operators such that, for $P$ given in eq:statwave, there exists a unique solution, $u ( x, \omega )\in \omega^{-1}\mathscr{H}_{\infty,\mathop{\mathrm{loc}}\nolimits}$, $( x, \omega ) \in \mathbb R^2$, of $P u = 0$ such that

Figures (2)

  • Figure 1: Comparisons of evolutions of a gaussian packet $g ( t - x )$, $t \ll 1$, $g (y ) = e^{-(x_0+y )^2) / \sigma - i \lambda (y- x_0))}$ , $\sigma = 0.05$, $\lambda = 10$, $x_0 = - 2$ for different values of the parameters in \ref{['eq:Bex']} (with $m = 1$) -- this corresponds to the model considered in horse. An animated version is available at https://math.berkeley.edu/ zworski/wave_multi_one.mp4. The code for producing this movie and the figure is enclosed in the appendix.
  • Figure 2: An illustration of Proposition \ref{['p:1Dwave']}: the solution to $( \partial_t^2 - \partial_x^2 + V ( x ) ) u = 0$, $\mathop{\mathrm{supp}}\nolimits V \subset [ - R , R ]$, with $u ( t, x ) = g ( t - x )$ for $t \ll -1$ is given by $g ( t - x ) + R_+ ( D) g ( t + x )$ for $x \leq - R$ and $T ( D ) g ( t - x)$ for $x \geq R$ (for all times; $t = 4$ shown). An animated version is available at https://math.berkeley.edu/ zworski/wave_pot.mp4.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof : Proof of Proposition \ref{['p:sim']}
  • Proposition 2
  • Lemma 3
  • proof
  • proof : Proof of Proposition \ref{['p:exu']}
  • Proposition 4
  • proof
  • ...and 8 more